The matrix algebra of changing coordinate system

The matrix algebra of changing coordinate system

Suppose we have the point $P(a,b)$ under the coordinate system $(1, 0), \; (0, 1)$, and we want to change the coordinate system to $(1/\sqrt{2}, -1/\sqrt{2}), \; (1/\sqrt{2}, 1/\sqrt{2})$. This is equivalent to solving the equation: \begin{align*} x \begin{pmatrix} 1/\sqrt{2} \\ -1/\sqrt{2} \end{pmatrix} +y \begin{pmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \end{pmatrix} &= a \begin{pmatrix} 1\\0 \end{pmatrix} +b \begin{pmatrix} 0\\1 \end{pmatrix} \end{align*} or, more concisely \begin{align*} \begin{pmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ -1/\sqrt{2} & 1/\sqrt{2} \end{pmatrix} \begin{pmatrix} x\\y \end{pmatrix} &= \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \begin{pmatrix} a\\b \end{pmatrix} \\ \begin{pmatrix} x\\y \end{pmatrix} &= \begin{pmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ -1/\sqrt{2} & 1/\sqrt{2} \end{pmatrix}^{-1} \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \begin{pmatrix} a\\b \end{pmatrix} \\ \end{align*}

Generally, if we wish to switch vector $v$ from system A to B, we do: \begin{align*} v_{B} &= B^{-1}Av_A \end{align*}